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In recent decades, a number of ways of dealing with causality in practice,such as propensity score matching, the PC algorithm and invariant causalprediction, have been introduced. Besides its interpretational appeal, thecausal model provides the best out-of-sample prediction guarantees. In thispaper, we study the identification of causal-like models from in-sample datathat provide out-of-sample risk guarantees when predicting a target variablefrom a set of covariates.  Whereas ordinary least squares provides the best in-sample risk with limitedout-of-sample guarantees, causal models have the best out-of-sample guaranteesbut achieve an inferior in-sample risk. By defining a trade-off of theseproperties, we introduce $\textit{causal regularization}$. As theregularization is increased, it provides estimators whose risk is more stableacross sub-samples at the cost of increasing their overall in-sample risk. Theincreased risk stability is shown to lead to out-of-sample risk guarantees. Weprovide finite sample risk bounds for all models and prove the adequacy ofcross-validation for attaining these bounds.

Causal Regularization: On the trade-off between in-sample risk and out-of-sample risk guarantees